Nuprl Lemma : fps-restrict-summation
∀[X:Type]
∀[eq:EqDecider(X)]. ∀[r:CRng]. ∀[f:PowerSeries(X;r)]. ∀[d:bag(X)].
(fps-restrict(eq;r;f;d) = Σ(x∈sub-bags(eq;d)). (f[x])*<x> ∈ PowerSeries(X;r))
supposing valueall-type(X)
Proof
Definitions occuring in Statement :
fps-restrict: fps-restrict(eq;r;f;d),
fps-scalar-mul: (c)*f,
fps-add: (f+g),
fps-single: <c>,
fps-zero: 0,
fps-coeff: f[b],
power-series: PowerSeries(X;r),
sub-bags: sub-bags(eq;bs),
bag-summation: Σ(x∈b). f[x],
bag: bag(T),
deq: EqDecider(T),
valueall-type: valueall-type(T),
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
lambda: λx.A[x],
universe: Type,
equal: s = t ∈ T,
crng: CRng
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
uimplies: b supposing a,
and: P ∧ Q,
cand: A c∧ B,
assoc: Assoc(T;op),
infix_ap: x f y,
squash: ↓T,
prop: ℙ,
true: True,
subtype_rel: A ⊆r B,
guard: {T},
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
implies: P ⇒ Q,
comm: Comm(T;op),
so_lambda: λ2x.t[x],
so_apply: x[s],
uiff: uiff(P;Q),
all: ∀x:A. B[x],
fps-single: <c>,
fps-coeff: f[b],
fps-scalar-mul: (c)*f,
fps-restrict: fps-restrict(eq;r;f;d),
crng: CRng,
rng: Rng,
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
ifthenelse: if b then t else f fi ,
power-series: PowerSeries(X;r),
bfalse: ff,
exists: ∃x:A. B[x],
or: P ∨ Q,
sq_type: SQType(T),
bnot: ¬bb,
assert: ↑b,
false: False,
not: ¬A,
ring_p: IsRing(T;plus;zero;neg;times;one),
group_p: IsGroup(T;op;id;inv),
bag-member: x ↓∈ bs,
rev_uimplies: rev_uimplies(P;Q),
top: Top,
bag-summation: Σ(x∈b). f[x],
bag-accum: bag-accum(v,x.f[v; x];init;bs),
list_accum: list_accum,
nil: [],
fps-zero: 0,
rng_zero: 0,
pi1: fst(t),
pi2: snd(t),
cons-bag: x.b,
monoid_p: IsMonoid(T;op;id),
ident: Ident(T;op;id),
fps-add: (f+g)
Lemmas referenced :
equal_wf,
squash_wf,
true_wf,
power-series_wf,
mon_assoc_fps,
fps-add_wf,
iff_weakening_equal,
fps-add-comm,
fps-ext,
fps-restrict_wf,
bag-summation_wf,
bag_wf,
fps-zero_wf,
fps-scalar-mul_wf,
fps-coeff_wf,
fps-single_wf,
sub-bags_wf,
crng_wf,
deq_wf,
valueall-type_wf,
rng_car_wf,
deq-sub-bag_wf,
bool_wf,
eqtt_to_assert,
assert-deq-sub-bag,
eqff_to_assert,
bool_cases_sqequal,
subtype_base_sq,
bool_subtype_base,
assert-bnot,
sub-bag_wf,
rng_zero_wf,
bag-summation-filter,
rng_plus_wf,
bag-eq_wf,
rng_plus_comm2,
crng_properties,
rng_properties,
rng_all_properties,
bag-extensionality-no-repeats,
decidable__equal_bag,
decidable-equal-deq,
bag-filter_wf,
subtype_rel_bag,
assert_wf,
single-bag_wf,
bag-filter-no-repeats,
sub-bags-no-repeats,
bag-single-no-repeats,
bag-member-single,
bag-member-filter,
assert-bag-eq,
bag-member_wf,
member-sub-bags,
and_wf,
bag-summation-single,
bag-summation-empty,
equal-empty-bag,
empty-bag-iff-no-member,
set_wf,
bag-member-filter-set,
rng_times_one,
rng_times_zero,
bag_to_squash_list,
list_induction,
list-subtype-bag,
subtype_rel_self,
list_wf,
nil_wf,
cons-bag-as-append,
bag-summation-append,
abmonoid_comm_fps,
mon_ident_fps,
isect_subtype_rel_trivial,
subtype_rel_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
sqequalRule,
applyEquality,
thin,
lambdaEquality,
sqequalHypSubstitution,
imageElimination,
extract_by_obid,
isectElimination,
hypothesisEquality,
equalityTransitivity,
hypothesis,
equalitySymmetry,
because_Cache,
cumulativity,
natural_numberEquality,
imageMemberEquality,
baseClosed,
universeEquality,
independent_isectElimination,
productElimination,
independent_functionElimination,
isect_memberEquality,
axiomEquality,
independent_pairFormation,
lambdaFormation,
setElimination,
rename,
unionElimination,
equalityElimination,
dependent_functionElimination,
dependent_pairFormation,
promote_hyp,
instantiate,
voidElimination,
hyp_replacement,
applyLambdaEquality,
setEquality,
dependent_set_memberEquality,
addLevel,
productEquality,
voidEquality,
equalityUniverse,
levelHypothesis,
independent_pairEquality,
functionEquality,
functionExtensionality
Latex:
\mforall{}[X:Type]
\mforall{}[eq:EqDecider(X)]. \mforall{}[r:CRng]. \mforall{}[f:PowerSeries(X;r)]. \mforall{}[d:bag(X)].
(fps-restrict(eq;r;f;d) = \mSigma{}(x\mmember{}sub-bags(eq;d)). (f[x])*<x>)
supposing valueall-type(X)
Date html generated:
2018_05_21-PM-10_10_37
Last ObjectModification:
2017_07_26-PM-06_34_24
Theory : power!series
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